ANTI HOPFIAN AND ANTI CO-HOPFIAN MODULES
University of Calgary, Canada.
In this talk, I will present some results I obtained on Anti hopfian and anti co-hopfian modules. These will appear in Contemporary Mathematics, AMS.
Hirano and Mogami refer to a module M as anti hopfian if M is not simple and every non-zero factor module of M is isomorphic to M. They completely characterize anti hopfian modules over rings possessing the property that cyclic modules are hopfian, by properties of the lattice L(M) of all sub modules of M and use this characterization to study the endomorphism ring S = End (M) of M.
Define a module M to be anti cohopfian if M is not simple and all non zero submodules of M are isomorphic to M. The proofs of Hirano and Mogami depend heavily on the fact that any non-zero module admits a subdirectly irreducible (equivqlently co-cyclic ) factor module. The dual notion to that of a co-cyclic module is that of a local module. In general it is not true that any non-zero module admits a local submodule. We completely characterize the rings satisfying the condition that any non-zero module admits a local sub module. We show that this class is closed under the formation of finite direct products but not under the formation of infinite direct products. We also show that semi-perfect rings belong to this class. Referring to a ring as coch if all co-cyclic modules are co-hopfian, we completely characterize anti co-hopfian modules M possessing a local sub module over a coch ring by properties of L(M) and use this characterization to study the ring S = End (M) of M. We also strengthen the results of Hirano and Mogami on anti hopfian modules. We construct several interesting examples and counter examples using modules over formal triangular matrix rings.